Canal classes and Cheeger sets

Giannopoulos, Hartzoulaki and Paouris asked in [A. Giannopoulos, M. Hartzoulaki and G. Paouris, On a local version of the Aleksandrov-Fenchel inequality for the quermassintegrals of a convex body, Proc. Amer. Math. Soc. 130 (2002) 2403–2412.] whether the best ratio between volume and surface area of convex bodies sharing a given orthogonal projection onto a fixed hyperplane is attained in the limit by a cylinder over the given projection. The answer to the question is known to be negative. In this paper, we prove a characterization of the positive answer in dimension [Formula: see text], using the Cheeger set of the common projection. A partial characterization is given in higher dimensions. We also prove that certain canal classes of convex bodies provide families of convex bodies satisfying a closely related inequality for a similar ratio.